Early on in the game, I chose to trust Draco's plan and go to the elves. But now I'm wondering what would happen if I trust Dora's lockpicking skills instead. Does that lead to an alternate storyline? And if so, is it a good one or a bad one?

Yes there is a different path. The two path join later, though there will be some differences depending on which path you took. Neither path is better, on Doras plan you'll get to know the human party members better instead of the elven members.